$ B = \left[\begin{array}{rr}3 & 5 \\ 5 & 2 \\ -2 & 3\end{array}\right]$ $ D = \left[\begin{array}{rr}2 & 3 \\ -2 & 5\end{array}\right]$ What is $ B D$ ?
Solution: Because $ B$ has dimensions $(3\times2)$ and $ D$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ B D = \left[\begin{array}{rr}{3} & {5} \\ {5} & {2} \\ \color{gray}{-2} & \color{gray}{3}\end{array}\right] \left[\begin{array}{rr}{2} & \color{#DF0030}{3} \\ {-2} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{2}+{5}\cdot{-2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{2}+{5}\cdot{-2} & ? \\ {5}\cdot{2}+{2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{2}+{5}\cdot{-2} & {3}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{5} \\ {5}\cdot{2}+{2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{2}+{5}\cdot{-2} & {3}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{5} \\ {5}\cdot{2}+{2}\cdot{-2} & {5}\cdot\color{#DF0030}{3}+{2}\cdot\color{#DF0030}{5} \\ \color{gray}{-2}\cdot{2}+\color{gray}{3}\cdot{-2} & \color{gray}{-2}\cdot\color{#DF0030}{3}+\color{gray}{3}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-4 & 34 \\ 6 & 25 \\ -10 & 9\end{array}\right] $